Introduction to Probability and Stochastic Processes

This is a first course in probability and stochastic
processes. It introduces the basic concepts of sample space,
probability, random variable, independence and conditional
probability, along with a variety of common useful distributions.
It begins with the intuitive idea of probability based on the
frequency of occurrence of an event, then presents the formal
mathematical definition of probability, and, eventually arcs back to
the intuitive idea with the so-called Law of Large Numbers which shows
that, indeed, frequency converges to probability in the appropriate
sense.

Under resources you find a link to a website called Random
which contains a lot of material in probability that can take you as
far as you wish. It also offers applets to run simple experiments. You
may use it to complement this material.

A broad strokes introduction with examples of the main topics
covered in the course is given in the following video.

Here is a more detailed list of the topics covered:

Counting Outcomes of Experiments and Sampling

Sample Spaces, Events, and Probability

Inclusion-Exclusion Principle

Monotone Sequences of Events

Conditional Probability

Bayes' Formula

Independence

The Gambler's Ruin Problem

Conditional Probability is a probability

Discrete Random Variables, Expectation, and Variance

Probability Mass and Cumulative Distribution Functions

Bernoulli, Binomial, and Poisson Distributions

Continuous Random Variables

Probability Density Functions, Expectation, and Variance

Uniform, Normal, and Exponential Distributions

Chebyshev's Inequality and the Law of Large Numbers

All important concepts and results will be introduced with and/or
complemented by numerous illustrative examples. The following is a
video overview.